Editing Hurwitz's automorphisms theorem (section)

== Interpretation in terms of hyperbolicity ==

One of the fundamental themes in [[differential geometry]] is a trichotomy between the [[Riemannian manifold]]s of positive, zero, and negative [[scalar curvature|curvature]] ''K''. It manifests itself in many diverse situations and on several levels. In the context of compact Riemann surfaces ''X'', via the Riemann [[uniformization theorem]], this can be seen as a distinction between the surfaces of different topologies:
* ''X'' a [[Riemann sphere|sphere]], a compact Riemann surface of [[genus (topology)|genus]] zero with ''K'' > 0;
* ''X'' a flat [[torus]], or an [[elliptic curve]], a Riemann surface of genus one with  ''K'' = 0;
* and ''X'' a [[Riemann surface#Hyperbolic Riemann surfaces|hyperbolic surface]], which has genus greater than one and ''K'' < 0.

While in the first two cases the surface ''X'' admits infinitely many conformal automorphisms (in fact, the conformal [[automorphism group]] is a complex [[Lie group]] of dimension three for a sphere and of dimension one for a torus), a hyperbolic Riemann surface only admits a discrete set of automorphisms. Hurwitz's theorem claims that in fact more is true: it provides a uniform bound on the order of the automorphism group as a function of the genus and characterizes those Riemann surfaces for which the bound is [[Mathematical jargon#sharp|sharp]].